#### Combinations and Mathematical Induction Model Question Paper

11th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. In an examination there are three multiple choice questions and each question has 5 choices. Number of ways in which a student can fail to get all answer correct is

(a)

125

(b)

124

(c)

64

(d)

63

2. The number of five digit telephone numbers having at least one of their digits repeated is

(a)

90000

(b)

10000

(c)

30240

(d)

69760

3. There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two points is

(a)

45

(b)

40

(c)

39

(d)

38

4. Number of all four digit numbers having different digits formed of the digits 1, 2, 3, 4 and 5 and divisible by 4 is

(a)

24

(b)

30

(c)

125

(d)

100

5. The product of r consecutive positive integers is divisible by

(a)

r!

(b)

r!+1

(c)

(r+1)

(d)

none of these

6. 5 x 2 = 10
7. Given four flags of different colours, how many different signals can be generated if each signal requires to use of 3 flags, one below the other?

8. How many three-digit odd numbers can be formed using the digits 0, 1, 2, 3, 4, 5? if The repetition of digits is allowed

9. Find the value of  $\frac { (n+3)! }{ (n+1)! }$

10. In how many ways a cricket team of eleven be chosen out of a batch of 15 players if there is no restriction on the selection?

11. Let p(n) be the statement "10n + 3" is prime. Show that p(2) is true but p(3) is not true.

12. 5 x 3 = 15
13. If (n + 1)C8 : (n - 3) P4 = 57:16, find n.

14. How many different words can be formed by using all the letters of the word "ALLAHABAD"? and in how many of them vowels occupy the even positions?

15. How many 'letter strings' together can be formed with the letters of the word "VOWELS" so that (i) the strings begin with E, (ii) the strings begin with E and end with W.

16. Out of 18 points in a plane, no three are in the same line except five points which are collinear. Find the number of lines that can be formed joining the points.

17. 32n - 1 is divisible by 8

18. 4 x 5 = 20
19. How many three-digit numbers, which are divisible by 5, can be formed using the digits 0, 1, 2, 3, 4, 5 if
(i) repetition of digits are not allowed?
(ii) repetition of digits are allowed?

20. How many strings can be formed using the letters of the word LOTUS if the word
(i) either starts with L or ends with S?
(ii) neither starts with L nor ends with S?

21. In how many ways 7 plus (+) signs and 5 minus (-) signs be arranged in a row so that no two minus signs are together?

22. Using principle of mathematical induction, prove that 41n -14n is a multiple of 27.